2025 Nov TZ3 P2 Q1

Kiran and Logan collect the following data about the river Afon, where \(x\) is the distance in metres from the source and \(y\) is the depth in centimetres.

This data is represented in the following scatter diagram.

Kiran knows that the depth of the river is \(0\) cm at the source.
Kiran calculates \(\bar{x}\) and \(\bar{y}\) for the seven points given in the table on page \(2\) and draws a line on the scatter diagram through the mean point \( (\bar{x}, \bar{y}) \) and the point \( (0,0)\).

1. Find

   [3]

   1. the value of \(\bar{x}\) and the value of \(\bar{y}\) ;

   2. the equation of Kiran's line.

For the seven points given in the table Logan finds the regression line of \(y\) on \(x\) with equation \(y = ax + b\), where \(a, b \in \mathbb{R}\).

2. Determine the value of \(a\) and the value of \(b\).

   [2]

3. By using the equation of Logan's regression line, estimate the depth of the river \(350\) m from its source.

   [2]